# Gauge invariance or global invariance, which one makes theory renormalizable?

We know that gauge theory is renormalizable, due to the Ward-Takahashi identity (for non-Abelian theory, it is Slavnov-Taylor identity), which reflects the conserved current of gauge symmetry.

But local (gauge) symmetry is not a real 'symmetry', since it cannot lead to a physical conserved current. When the gauge group is non-Abelian, local gauge invariance can lead to either a gauge-invariant but non-conserved current, or a gauge-dependent but conserved current (for $U(1)$ group these two currents coincide). But global symmetry, leads to a physical (global) invariant conserved current (for non-Abelian group, gauge field transform under global transformation, too), and this can lead to corresponding Ward-Takahashi identity.

Now here is my question, if a gauge theory is global but not local invariant, does it renormalizable? Specifically, if in SM Lagrangian we change Higgs covariant differential $D_\mu$ to ordinary differential $\partial_\mu$, does the theory renormalizable? If the change is done, then the Yukawa interaction term destroys the local $SU(2)\times U(1)$ symmetry, but conserves the global one.

• And the question I have is, if there is a covariant differential $D_\mu$ for fermion, but a ordinary differential $\partial_\mu$ for Higgs, will this Lagrangian renormalizable? The model is invariant under global $SU(2)\times U(1)$ symmetry, but non-invariant under the local one (Yukawa term breaks the symmetry). Oct 3, 2016 at 8:22